Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Prove proposition on real numbers and inverses.

Prove the following proposition Let $x, y \in \mathbb{ R}>0$. If $x < y$ then $0 < y^{-1 }< x^{-1}.$ So far I've gotten that since $x, y > 0$ then $x^{-1}, y^{-1} > 0$.
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1answer
66 views

Linear Algebra - Find inverse of $A$

I have this problem : $$A = \left(\begin{array}{ccc} 3 & -1 & 1 \\ 2 & 0 & 1 \\ 1 & -1 & 2 \end{array}\right) $$ 1) Show that $A^3-5A^2+8A-4I=0$. 2) Using (1) To find ...
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16 views

Solving simultaneous equations with matrices

I have a Matrix $B = \begin{pmatrix}2&1\\3&5\end{pmatrix}$ and its inverse $B^{-1}=\frac17\begin{pmatrix}5&-1\\-3&2\end{pmatrix}$ I also have a set of simultaneous equations: ...
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5answers
53 views

True or false? Prove it.

If $A$ is an $n\times n$ invertible matrix and $B$ is an $n\times m$ matrix, then $\operatorname{rank}(AB) = \operatorname{rank}(B)$. Is this true or false? I've tried proven that if $B=0$, then ...
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1answer
25 views

What properties do I have if I know $f$ and $f^{-1}$inverse are differentiable?

My goal is to show that $(f^{-1})'(y) = 1/[f'(f^{-1}(y)]$ for all $y$ in $(a,b)$. I have no idea where to start. I know that $f^{-1}$ and $f$ are differentiable.
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7answers
141 views

$f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$?

Give two functions $f$ and $g$ s.t. $$f \circ g = \operatorname{id}$$ but $$g \circ f \neq \operatorname{id}$$ or a proof that this is impossible. This must be trivial, but I can't figure it out :) ...
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2answers
28 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...
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4answers
60 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
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0answers
38 views

Inverse Relation of Irreflexive Property.

We are taking the inverse of relation to check that inverse of R is transitive, reflexive , symmetric and anti-symmetric to as it is on R (not inverse).. My question is that why we are not taking the ...
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0answers
34 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
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1answer
52 views

Is this notation for inverse functions bad?

I'm trying to find useful notation for inverse functions that isn't too much in conflict with other notation already in use, but I'm wondering if this notation will come back and bite me in the ...
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0answers
20 views

Compute new inverse when old inverse and new and old matrix known

Say I have a matrix $M$ and know its inverse $M^{-1}$. Then every element changes so that $M'=M+(M'-M)$. Is there a fast way to find $M'^{-1}$ from this information? That is without computing the new ...
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1answer
46 views

The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
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1answer
27 views

The relation of domain and image of a function and its inverse

Theorem: Let both $f$ and $f^{-1}$ be functions. $\newcommand{\dom}{\operatorname{dom}}\newcommand{\im}{\operatorname{im}}$ Then $\dom(f) = \im(f^{-1})$ and $\dom(f^{-1}) = \im(f)$. Let $f: X ...
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1answer
18 views

Inverse of a function involving a Jacobian.

Why is it true that if the inverse of both $ \tilde{f} $ and $ f $ exists then: $$ \tilde{f}\left(\vec{x}\right) = [Df(x_{0})]^{-1} f(\vec{x}) $$ $$ \implies \tilde{f}^{-1}(\vec{x}) = ...
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0answers
60 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
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1answer
25 views

How to verify an algebraic structure is a ring

I have a problem which ask me to verify that to structures are rings. However, I'm unsure of how exactly to check each property. I believe that the first is closed but not sure how to check the ...
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1answer
18 views

Linear maps, inverses and associated matrices?

This is likely a very simple question but if we have a linear map $f$ with an associated matrix $A$ is it a necessary and sufficient condition that for $f$ to have an inverse then $A$ must also have ...
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4answers
67 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
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0answers
14 views

Using the inverse of the matrix find all the solutions of the following systems of equations?

I found the inverse using row operations and the identity matrix but I dont know where to go from here. Can someone direct me please ?
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1answer
48 views

What are the possible values of $x$?

For what values of $x$ does this equation holds? $$2\arctan(x)=\arctan\left(\frac{2x}{1-x^2}\right)$$ The answer is $-1<x<1$ Why? How can we say this?
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1answer
29 views

Inverse matrix as a sum of matrix powers [duplicate]

I have matrix $ A\in \mathbb{C}^{n x n}$ and $A$ is invertible. How can I show that coefficients $c_0,...,c_{n-1}$ exist : $A^{-1} = c_0I+c_1A+...+c_{n-1}A^{n-1}$ I tried to solve it first by ...
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35 views
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27 views

Proofs with algebraic structures (rings)

If one is given a ring $R$ with a unity $u$, what are the steps one would have to take to prove that some element of $R$ named $s$ has a multiplicative inverse, where $-s$ also has a multiplicative ...
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1answer
21 views

Polynomial has right inverse implies invertible?

If $p:\Bbb R\rightarrow \Bbb R$ is a real polynomial such that $p$ has a right inverse $q$, does it follow that $p$ is invertible? That is, must $q$ also be a left inverse of $p$? The question ...
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1answer
21 views

Two roots of $\arcsin(x)$ in the range $[0,2 \pi]$

I am baffled with how to write the two roots of arcSin$(x)$ in the range $[0,2 \pi]$, while $x \in [-1,1]$, such that one root can be directly calculated in terms of the other root. For instance, we ...
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1answer
40 views

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$.

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$. Proof: Let $f(x) = \sin x$, for $x$ in $(-1,1)$. Then let $x_{0}$ be in (-1,1). Then $f'(x_{0})$ = ...
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1answer
21 views

Fourier Transform Inverse of 1 / (jw - a)

I want to find the inverse fourier transform of $$ \frac 1 {j \omega - 1} $$ The fourier transform of $$ e^{-at} u(t) $$ is $$ \frac {1}{j \omega + a} $$ This result if true ONLY if a > 0. If a ...
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30 views

Differential operator and its inverse

suppose we have a differential operator $D$ in terms of a variable $x$ and its inverse is denoted by $D^{-1}$ , then is it possible that $DD^{-1}=\delta (x)$ or $D^{-1}D=\delta (x)$? If so, then what ...
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48 views

Inverse Square law problem, how to calculate for distances.

i've got a bit of a problem with the inverse square law (I1/I2=D2 squared/D1 squared)(Where I=intensity and D=distance) I need to change a distance from 1000mm to 400mm (I'm a Radiographer). Most of ...
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1answer
50 views

Catch 22 situation involving inverting a function and finding the range of the function.

Let $f(x) = \sqrt{x+5} - \sqrt{x-5}$ Calculating the inverse: $y = \sqrt{x+5} - \sqrt{x-5}$ $y + \sqrt{x-5} = \sqrt{x+5}$ $y^2 + x - 5 + 2y\sqrt{x-5} = x + 5$ $\frac{(10 - y^2)^2}{4y^2} + 5 = x$ ...
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1answer
26 views

Inverse of a particular operator

I need help finding the inverse of the following operator. I am not sure about how to start. Any help would be hugely appreciated. Operator: $( I + \frac{\partial^2}{\partial x^2})$ Edit: I ...
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1answer
59 views

Finding derivative of the inverse without the inverse

We are given a function $$f(x)=4\arcsin(\sqrt{x})+2\arcsin(\sqrt{1-x})$$ The derivative of $f$ is: $$f'(x)=\frac{1}{\sqrt{x-x^2}}$$ I would like to find the maximum value of $f^{-1}$. I think I have a ...
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1answer
32 views

quadratic equation modulo some number

I read a post that $$ax^2+bx+c \equiv 1 \pmod p$$ can be solved in a similar way we solve a simple quadratic equation, just by replacing division by $2a$ by modulo inverse of $2a$ and square root of ...
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1answer
50 views

Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$ for $f(x) = x^2e^{x^2}$.

Let $f(x) = x^2e^{x^2}$, and assume that $(e^x)' = e^x$ for all $x$ in $R$. a) Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$. Proof: Suppose that $f(x) = x^2e^{x^2}$, then finding ...
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1answer
21 views

right inverse and supplement of kernel in a banach

For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that : $Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right ...
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1answer
17 views

Question Regarding Inverses In a Function

Here is my current issue. Our teacher asked a question related to the finding of an inverse of 2. Here is all of the given information: Define "a cross b" as such: a ☢ b = ab + (a + b). Use this ...
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1answer
23 views

Inversion of a symmetric and positive definite matrix with or without a column and row

Suppose to have a symmetric and positive definite matrix $\boldsymbol{\Sigma}$ and suppose to know its inverse $\boldsymbol{\Sigma}^{-1}$. Let $\boldsymbol{\Sigma}_{+}= \left( \begin{array}{cc} ...
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3answers
28 views

Inverse Trig Functions, finding Domain and Range

I understand the restricted domains of inverse trig functions, but what about: I don't quite understand how to find the domain and range of this function.
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1answer
110 views

Moore-Penrose Pseudo-inverse of a matrix on adding 1 new row/column

Given that I know the pseudo-inverse of a matrix(not necessarily a square matrix), how to calculate the pseudo-inverse of the matrix I get by adding a single row/column to the original matrix? i.e, ...
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1answer
37 views

Inverse of the matrix product $\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T}$

If I have an $n\times n$ symmetric matrix $\boldsymbol{S}$ and a $m\times n$ matrix $\boldsymbol{A}$ is there any relation between $(\boldsymbol{A} \cdot\boldsymbol{S} \cdot \boldsymbol{A}^{T})^{-1}$ ...
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387 views

Inverse of sparse matrix is not generally sparse

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the ...
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1answer
39 views

Find Invertible and NonInvertible Matrix

Can someone help me to understand this problem? I don't know where to begin. Find an invertible matrix $A$ and a noninvertible matrix $B$ both of which satisfy $$M^2=3M$$ Thanks, Rusty
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Solution for set of matrix equations involving an inverse

I am encountering the following set of three matrix equations for which I search a solution in terms of ${\bf M}\in\mathbb{R}^{N\times N}$ and ${\bf D}\in\mathbb{R}^{Q\times N}$, $${\bf M}{\bf W} = ...
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1answer
56 views

Find all right inverses of matrix A.

I'm given the matrix A where it's first row is $(2, -1, 3)$ and second row is $(1, 2, 1)$ and I'm told to find all the right inverses of it. First I tried doing A times a 3x2 vector B (just a vector ...
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2answers
88 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Statement. $I-A$ ...
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2answers
64 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
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1answer
87 views

Simple formula for a sieries like 1, 2, 5, 10, 20, 50, 100, …

I'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f(4) ...
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1answer
57 views

Solve Inverse Linear Congruence

I want to solve Linear congrunece : 9x+2 ≡ 6(mod 1453) using inverse of 9 mod 1453. Inverse of 9 mod 1453 is 323. Now to solve it I subtract 2 from left and right side which gives me 9x ≡ 4(mod 1453), ...
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3answers
81 views

Bijection, and finding the inverse function

I am new to discrete mathematics, and this was one of the question that the prof gave out. I am bit lost in this, since I never encountered discrete mathematics before. What do I need to do to prove ...